| L(s) = 1 | − 16·5-s − 49·7-s − 8·11-s + 684·13-s + 2.21e3·17-s − 2.69e3·19-s − 3.34e3·23-s − 2.86e3·25-s + 3.25e3·29-s + 4.78e3·31-s + 784·35-s − 1.14e4·37-s − 1.33e4·41-s − 928·43-s − 1.21e3·47-s + 2.40e3·49-s − 1.31e4·53-s + 128·55-s − 3.47e4·59-s − 1.03e3·61-s − 1.09e4·65-s + 1.01e4·67-s − 6.27e4·71-s − 1.89e4·73-s + 392·77-s + 1.14e4·79-s − 8.89e4·83-s + ⋯ |
| L(s) = 1 | − 0.286·5-s − 0.377·7-s − 0.0199·11-s + 1.12·13-s + 1.86·17-s − 1.71·19-s − 1.31·23-s − 0.918·25-s + 0.718·29-s + 0.894·31-s + 0.108·35-s − 1.37·37-s − 1.24·41-s − 0.0765·43-s − 0.0800·47-s + 1/7·49-s − 0.641·53-s + 0.00570·55-s − 1.29·59-s − 0.0355·61-s − 0.321·65-s + 0.275·67-s − 1.47·71-s − 0.415·73-s + 0.00753·77-s + 0.205·79-s − 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 16 T + p^{5} T^{2} \) |
| 11 | \( 1 + 8 T + p^{5} T^{2} \) |
| 13 | \( 1 - 684 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2218 T + p^{5} T^{2} \) |
| 19 | \( 1 + 142 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 3344 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3254 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4788 T + p^{5} T^{2} \) |
| 37 | \( 1 + 310 p T + p^{5} T^{2} \) |
| 41 | \( 1 + 13350 T + p^{5} T^{2} \) |
| 43 | \( 1 + 928 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1212 T + p^{5} T^{2} \) |
| 53 | \( 1 + 13110 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34702 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1032 T + p^{5} T^{2} \) |
| 67 | \( 1 - 10108 T + p^{5} T^{2} \) |
| 71 | \( 1 + 62720 T + p^{5} T^{2} \) |
| 73 | \( 1 + 18926 T + p^{5} T^{2} \) |
| 79 | \( 1 - 11400 T + p^{5} T^{2} \) |
| 83 | \( 1 + 88958 T + p^{5} T^{2} \) |
| 89 | \( 1 + 19722 T + p^{5} T^{2} \) |
| 97 | \( 1 - 17062 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.55459690028161290337774082771, −9.937316717899621250490684495683, −8.575112552914830418078523997200, −7.929367140853306396169152744262, −6.55311814864751536479426054348, −5.73197414798066843777944272257, −4.21717783589933081406849014672, −3.23641400235369632726176733783, −1.58117296822369444835957940989, 0,
1.58117296822369444835957940989, 3.23641400235369632726176733783, 4.21717783589933081406849014672, 5.73197414798066843777944272257, 6.55311814864751536479426054348, 7.929367140853306396169152744262, 8.575112552914830418078523997200, 9.937316717899621250490684495683, 10.55459690028161290337774082771
